Coherent atomic oscillations and resonances between coupled Bose-Einstein condensates with time-dependent trapping potential

Coherent atomic oscillations and resonances between coupled Bose-Einstein condensates

with time-dependent trapping potential

F. Kh. Abdullaev

Physical-Technical Institute, Uzbek Academy of Sciences, G. Mavlyanov Street, 2-b, 700084 Tashkent-84, Uzbekistan and Instituto de Fı´sica Teorica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, Brazil

R. A. Kraenkel

Instituto de Fı´sica Teorica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, Brazil ~Received 1 March 2000; published 20 July 2000!

We study the quantum coherent tunneling between two Bose-Einstein condensates separated through an oscillating trap potential. The cases of slow and rapid varying in the time trap potential are considered. In the case of a slowly varying trap, we study the nonlinear resonances and chaos in the oscillations of the relative atomic population. Using the Melnikov function approach, we find the conditions for chaotic macroscopic quantum-tunneling phenomena to exist. Criteria for the onset of chaos are also given. We find the values of frequency and modulation amplitude which lead to chaos on oscillations in the relative population, for any given damping and the nonlinear atomic interaction. In the case of a rapidly varying trap, we use the multiscale expansion method in the parameter «51/V, where V is the frequency of modulations, and we derive the averaged system of equations for the modes. The analysis of this system shows that new macroscopic quantum self-trapping regions, in comparison with the constant trap case, exist.

PACS number~s!: 05.30.Jp

I. INTRODUCTION

It has recently been shown that there exists a macroscopic quantum phase difference in processes connected with atomic waves. Namely, this effect is observed between two tunnel coupled Bose-Einstein condensates @1–3#. The cou-pling occurs due to a trap potential which has the form of a double-well potential. The theoretical investigation shows that there is possibly an interesting phenomenon of periodic oscillations of the atomic population between condensates and quantum self-trapping of population in the dependence of the relative phase between condensates@4–8#. Analogous phenomena which have been studied are the ac Josephson effect@9#and the periodic exchange of power and switching of electromagnetic waves between cores in nonlinear optical couplers@10#.

In particular, there is a direct analogy between the tunnel-ing phenomena in two prolongated Bose-Einstein conden-sates and two tunnel-coupled single-mode optical fibers. In this optical analogy, the role of the chemical potential is played by the propagation constant, and the nonlinear inter-action between atoms is analogous to the Kerr nonlinearity of optical media. The tunnel coupling arising from the over-laps of the electromagnetic fields outside of dielectric cil-inders ~fibers! @11–13# exactly corresponds to the tunnel coupling between two Bose-Einstein condensates due to the overlaps of the wave functions.

In this context, it is natural to investigate the influence of a time-varying coupling on the quantum coherent tunneling process. In the nonlinear optical coupler analogy, this corre-sponds to the variation in the longitudinal direction coupling @14,15#. As was shown recently, the variation in time of the trap potential can lead to resonant oscillations of the Bose-Einstein condensate @16,17#. The numerical solution of the Gross-Pitaevsky equation shows that the rapid variation in

time of the trap potential can lead to a bifurcation of the effective ~averaged! form of the trap potential and conse-quently to splitting of the BEC @18#.

From what we have said above, we could wait for new phenomena in the process of quantum coherent atomic tun-neling ~QCAT! too. Recalling the analogy with Josephson effects in coupled junctions, we note that in the last case it is very difficult to produce such effects, the difficulty coming from the fact that it is hard to implement the time-varying overlap properties of junctions.

Thus, in this paper we will study the atomic tunneling between two tunnel-coupled BEC’s in a double-well time-dependenttrap. Namely, in Sec. II we will study the nonlin-ear resonance phenomena in the interference and, in particu-lar, the stationary regimes, with damping taken into account. The phase and population damping are considered as well. The phase-locked states are interesting because of the sup-pression in these states of the fluctuations of the relative phase between two condensates. The regions of regular and chaotic oscillations of the relative atomic population are cal-culated in Sec. III, using the Melnikov function approach. In Sec. IV the macroscopic quantum interference in two over-lapped condensates with rapidly varying traps is investi-gated. We derive the system of averaged equations for coupled modes and define in Sec. V new regimes of param-eters for the macroscopic quantum self-trapping ~MQST!.

II. FORMULATION OF PROBLEM. SLOWLY VARYING IN TIME TRAP

atomic scattering length. The attractive case can be obtained analogously, using symmetries of the equation. This sys-tem of equations is valid in the approximation of a weak link between condensates. Comparison with the numerical solu-tion of GPE shows that the system ~1! is a reasonable ap-proximation for z<0.5– 0.6. For the strongly overlapped condensates, the system should be modified. For a periodi-cally varying K(t)₅K01K1sin(Vt) we have limitations on the parametersK1 andV for which the two-mode approxi-mation is valid. The small deforapproxi-mation of bound states re-quiresK1!K0. To avoid the resonances of the modulations with normal modes of the spherical well, we should require that the difference of energy between the ground state and the first normal mode be larger than the modulation energy, i.e.,dE@\V. Taking into account thatdE;\v0@7#, where v0 is the harmonic-oscillator frequency, we find the condi-tion V_!v0. Taking into account that v0@vL52K0—the frequency of the linear oscillations of the atomic population, defined by the tunneling frequency—we conclude that by a proper choice of parameters we can have the resonant case V.vL or the rapidly varying modulationV@vL.

Introducing new variables u_i5

A

N_iexp(iu_i), z5(N1 2N2)/NT,NT5N11N2, andc5u12u2, whereNi, andui

are, respectively, the number of atoms and phases in the ith trap, we get the following system:

zt522K~t!

A1

2z2sinF2hFt, ~2!

F_t5nLz12K~t!z

A1

₂z2cosF1DE~t!. ~3!

where L₅(a11a2)NT and n561 for the positive and

negative atomic scattering length, respectively. In what fol-lows, we will consider the case n51 (as.0), if not stated

otherwise. We have included in Eq. ~2! the damping term

hf_t(t). It appears if we take into account a noncoherent dissipative current of normal-state atoms, proportional to the chemical potential differenceDm. For the other type of over-lapping condensate, there might be another type of damping. For example, for the two interacting condensates with differ-ent hyperfine levels in a single harmonic trap, the damping has the form hz(t).

The Hamiltonian of the unperturbed system @K(t)

5const,DE5const,h50#is

H5Lz 2

2 22K

A1

2z

2_{cos~ F !}

1DEz. ~4!

D

¯_t52V₁DE1Lz2 K1z

A1

₂¯z2sinD

¯_. ~6!

Let us now look for the fixed points of Eq.~5!. Introduce first the notation a5V₂DE. From the system~5!we find the following equation for the fixed points:

zc5

S

12

h2_V2

K₁2cosD_c2

D

1/2

, ~7!

a₅zc

S

l2K12

sin 2D_c

2hV

D

. ~8!

For small z_c, z_c2_!1, we can find the explicit solution

D_c5arccos

S

hV

K1

D

,

~9!

zc'

V₂DE

L₂

A

K₁2₂h2_V2.

The existence of a stationary solution is connected with the circumstance that the damping of the oscillation in quan-tum tunneling is compensated by the periodic variation of the trap. In Ref.@6#the influence of the damping on the macro-scopic quantum interference has been investigated numeri-cally and degradation of the interference due to damping has been analyzed. The predicted stationary solutions, where these effects are compensated by the variable driving, are indeed phase-locked solutions. Thus we can think that the fluctuations of the relative phase will be suppressed in such states. It is an important prediction since BEC’s have phase fluctuations and it is difficult to maintain the constant relative phase of condensates. This phenomenon can be useful to develop the phase standard for the Bose-Einstein condensate—a problem attracting considerable attention re-cently@19#.

We can estimate the value of fixed points for an experi-mental situation. For example, when V₅0.7, h₅0.1, K1 50.2, and L₅2.5, we obtain the fixed points z1c50.2599

The same system of equations also describes the nonlinear Josephson-type oscillations in the relative population of a driven, component Bose-Einstein condensate. The two-component condensates are those in which two different hy-perfine states can be populated and confined in the same trap. A weak driving field couples two internal states and as a result the oscillations in the relative population will occur @20#. In this case the damping term has the form G₅₂hz. Repeating the above-mentioned procedure, we obtain the av-eraged system

z

¯_t52K1

A1

2¯z2cosD¯2h¯z,

~10!

D

¯

t5L¯z1DE2V2 K₁¯z

A1

₂¯z2sinD ¯_.

In the model@20#,DE corresponds tom22m12k, where m_i are chemical potentials and k is frequency detuning k 5vd2v0, v0 is the separation in frequency of two hyper-fine states, andv₀is the frequency of drive. The fixed points are

D¯

c5arccos

S

hzc

K1

A1

2zc

2

D

,

~11!

Lzc2a2 K1zc

A1

₂z_c2

S

12

h2_z

c

2

K₁2~12z_c2!

D

1/2

.

For the case of smallz_c2_!1 we have the estimate forz_c,

z_c5D_LE2V 2K1 1

~ DE2V !3_~K 1 2

2h2_! 2K1~ L2K1!4

. ~12!

The typical values of the parameters are V₅0.7, m2 2m12k5DE51, h50.1, K150.2, and L54, and the fixed points arez3c520.111 andz4c520.13. The analysis

shows that the first fixed point is stable and second is un-stable.

In Fig. 2 we present the plot of solution of the full system

for the damping term hz with the above-mentioned set of parameters. All solutions decay to the first fixed point.

III. CHAOTIC OSCILLATIONS IN RELATIVE ATOMIC POPULATION

In this section we consider the interference in the system which in some limit is mathematically equivalent to the mo-tion of a particle in a double-well potential under parametric periodic perturbations. The particle motion is of course un-harmonic and the nonlinear resonances between oscillations of a particle and oscillations of the trap potential are pos-sible. One of consequences of this is the possible appearance of chaotic dynamics in such systems. We will now study the conditions for the appearance of the chaotic dynamics in atomic tunneling.

To proceed, it will be useful first to consider some aspects of the unperturbed system, that is, when the coupling is con-stant. In this case the system~2!is equivalent to the quartic ~Duffing!oscillator@6,23#.

We can write the equation forz(t) as

ztt52

]V

]z, ~13!

where the potential V(z) is given by

V~z!₅z2~a₁bz2!, a₅2K₀2₂LH

2 , b5 L2

8 , ~14!

and His the Hamiltonian. The total energy of the effective particle is

E₀₅zt

2

2 1V~z!52K 2

2H2/2.

When E0.0, we have the oscillating regime with

^

z(t)

&

50 ~nonlinear Rabi oscillations!. This case corresponds to the periodic flux of atoms from one BEC to the other. When

E0,0, the motion of the effective particle is confined in that of the wells. This case corresponds to the localization of atomic population in one of the condensates—the so-called macroscopic quantum self-trapping ~MQST!. The case E0 FIG. 1. The typical behavior of the solutions to Eq.~2!when the

damping is taken in the form G5hft. This case corresponds to V50.7,h50.1,K₁50.2, andL52.5. The theoretical value isz_c

50.2599, the numerical value isz_c50.26.

FIG. 2. The typical behavior of the solutions to Eq.~2!when the damping is taken in the formG5hz. This case corresponds toV

~15!

sin2~ Fs!5

s2_sech2_~

_A2

_at!tanh2_~

_A2

_at!

2K₀2@b2asech2~

A2

at!# ,

whereH5Hs52K.

The motion near the separatrix is very sensitive to the change of the initial condition and a stochastic layer appears, leading to the chaotic dynamics of the solutions of the sys-tem ~2!. The Melnikov function method is effective to find the regions of chaotic behavior@21,22#. The system~2!may written in the form r_t5f₁eg and has a hyperbolic fixed point. According to this method, we need to calculate the Melnikov function M(t0),

M~t0!5

E

2`

`

@f2sg1s~t,t0!2f1sg2s~t,t0!#dt. ~16!

As is well known, the existence of zeros ofM(t0) indicates the intersection of separatrices and the existence of ho-moclinical chaos. Substituting the expressions for f_i,g_ifrom the system~2!and calculating the integrals, we find the final expressions for the Melnikov function. We shall give the results for two possible damping terms.

A. The case of the damping in the formhFt

The integrations in Eq.~16!are cumbersome, and will be omitted. The final result for the Melnikov function is

M1~t0!52K1F1cos~ Vt0!2F2h, ~17!

where

F15

pLV2

4K0bsinh

S

pV

2

A2

a

D

,

~18!

F25

2L

A2

a

A

ab2a2~ L22Hs!arctan@

A

a/~b2a!#

2

A2

a

16~b2a!b2G1,

where

G15@22b2~2Hs2L !~bHs24aL13bL !#

3arctan~a/Aab2a 2_!

A

ab₂a2

22@4a2L2

25abL2

12b2~4H_s224HsL13L2!#.

~19!

The simple zero of M(t0) is absent if the condition

h_.F1

F2

~20!

is satisfied. In Fig. 3 we plot this criterion in the (K1,V) plane for h50.1. Regions below the curve correspond to regular oscillations and those above correspond to chaotic oscillations ofz(t). The criterion gives a good lower bound on the region of chaos in this plane. We verified this lower bound in the (K₁,V) plane by numerical simulations and found a good agreement with the formulas~20!.

When the damping coefficient is zero, we have the ex-pression for the width of stochastic layer near the separatrix as

FIG. 3. The regions of regular and chaotic oscillations in plane

K1, Vfrom the Melnikov approach.

FIG. 4. The width of stochastic layer as a function of frequency V, when h50, K051, K150.2, L59.9, z(0)50.6, and F(0)

D5 pLK1V 2

4K0bsinh

S

pV

2

A2

a

D

. ~21!

In Fig. 4 we plot the width of the stochastic layer as a function of the frequency for K150.2, L59.9, z(0)50.6, and F(0)₅0. The maximum of the width is given by the solution of the equation

Vtanh

S

pV 2A2a

D

5

4

A2

a

p .

For this choice of parameters, we have V_m51.89. For V >2,a>1 we get the estimateV_m'4

A2

a/p. It can be seen that, for the high frequenciesV_@

A

a, the stochastic layer is exponentially narrow and motion is regular. The analysis of this case will be performed in detail in the next section. In Fig. 5 we plot the typical oscillations in the relative atomic population for L₅9.9, K051, K150.2, z(0)50.6, and F(0)₅0 forV₅1.5. It turns out that forVbetween 1.5 and 3.22 we have chaotic oscillations, and for small or largeV, regular oscillations. In Fig. 6 the phase portrait is given. In Fig. 7 the influence of the damping on chaos is illustrated. Parameters are the same as in Fig. 5 and h₅0.15. In this case the oscillations become regular.

B. The case of the damping term in the formhz„t… The calculation of the integrals which give the Melnikov function is again cumbersome, and one finds the following result:

M~t0!5F1cos~ Vt0!1hF3, ~22!

where

F35

L₂2H_s

A2

~b₂a!arctan@

A

a/~b2a!#1 3L

A

a

2_&b . ~23!

The zeros of M(t0) are absent when the condition

h_.F1

F3

~24!

is satisfied.

If we consider for example the case when K150.2, V 50.7, L₅2.5,z(0)₅0.6, and F(0)₅p, the damping con-stant should be h50.15. In Figs. 7 and 8 the chaotic and periodic oscillations of the relative population in the depen-dence of the frequency for fixedz(0),F(0) are plotted. FIG. 5. The typical oscillations of the relative populationz(t)

forK₁₅1.2,L59.9,z(0)50.6,F050, andV51.5.

FIG. 6. The phase portrait in the (F,z) plane for the same values of the parameters as in Fig. 5.

FIG. 7. The typical oscillations of the relative populationz(t) forK150.2,L52.5,z(0)50.9,F05p, andV53.22.

where z5t/e, e!1, and t_k5ek_{t are slow times. The}

aver-ages over a period of rapid oscillations are

^

ui

&

5

^

ui

&

50.

The functionsui,v_iare assumed to be functions of the rapid

timezand the slow varying functions U,V. The derivatives can be written in the form

dU dt 5

dU dt01

edU

dt11

¯ ~26!

and analogously for the derivative ofV.

Below, we will assume E15E2. Then the time depen-dence in E(t) can be removed by a simple field transforma-tionu(v)5u(v)exp„(i/h)*E(t

8

)dt

8

…. In the zeroth order in

ewe get the equations

i\

S

]u0

]t01

]u1

]z

D

5Eu02K~z!v₀1auu₀u2u₀, ~27!

i\

S

]

v₀

]t01

]u1

]z

D

5Ev02K~z!u01auv₀u2v₀. ~28!

Requiring the exclusion of the secular inzterms, we find the correctionsu1,v₁,

u₁₅i

\v0~m12

^

m1

&

!, ~29!

v₁5

i

\u0~m12

^

m1

&

!, ~30! wherem15*0

z

@K(s)2

^

K

&

#ds and the system foru0,v₀

be-comes

i\]u0

]t05

Eu₀₂

^

K

&

v₀1auu₀u2u₀, ~31!

i\]

v₀

]t05

Ev₀₂

^

K

&

u₀₁auv₀u2v₀. ~32!

In the order ewe have

]u₀

]t15

0, ]

v₀

]t15

0, ~33!

and we obtain, for the second-order corrections u2,v₂; the

elimination of the secular terms result in

m₂₅

E

0 z

„m₁~s!₂

^

m₁

&

…ds, M₅~

^

m₁2

_&

2

^

m1

&

2!. From this order we get the equations

]u0

]t252

2i Ma

h3 @2uv0u

2_u 02v

0 2_u

0

*#, ~36!

]v₀

]t₂52

2iaM

h3 @2uu0u2v₀2u

0 2

v

0

*#. ~37!

Finally, we have in the next order,O(e2_{), the averaged} sys-tem

i\]u0

]t 2Eu02auu0u

2_u 0

52

^

K

&

u012

e2_aM

\2 @2uv0u2u02v

0 2

u₀*#, ~38!

i\]u0

]t 2Ev02auv0u

2_v 0

52

^

K

&

v₀12

e2_aM

\2 @2uu0u2v₀2u

0 2_v

0

*#. ~39!

For periodic modulations of tunnel coupling, we obtain M

5K₁2/V2.

This averaged system is the main result of this section. We see that, in comparison with the constant coupling case, new effects appear. First, the cross term appears, correspond-ing to a change of type in the effective nonlinearity in coupled BEC dynamics. Second, there appears an additional

nonlinear phase-sensitive coupling. This term leads to new minima in the effective potential picture. In the constant cou-pling case, the oscillations ofz(t) are described by the Duf-fing oscillator. Now we have a more complicated nonlinear oscillator, describing more complicated interference patterns.

V. ANALYSIS OF THE AVERAGED DYNAMICS

Using the variables introduced in Sec. II, we get the sys-tem

F_t5Lz12

^

K

&

z

A1

₂z2cos~ F !2ndz@22cos~2F !#, ~41!

where we have defined d5e2_aM/h2_{. This system has the} following Hamiltonian:

H5L1z 2

2 22

^

K

&

A1

2z

2_{cos~ F !}

2d₂~1₂z2!cos~2F !, ~42! whereL₁₅L₂2d.

When the parameter d50, we get back the system ~2!. The effective particle motion is that of the double nonrigid pendulum system.

Let us consider the different regimes of oscillations.

A. Linear regime

This is the case whereuzu!1,uFu!1. Then, from the sys-tem ~40!, it follows that

z_tt522~

^

K

&

1d!~ L₁2

^

K

&

2d!z.

Then the frequency of linear oscillations of the number of atoms is

v_L2522~

^

K

&

1d!~ L12

^

K

&

2d!.

Thus, the frequency of the linear oscillations grows with

dv2

5v_L22v₀252d(

^

K

&

1L₂d), in comparison with a nonmodulated case.

B.z2_{™1, and}Fis not small

The phase dynamics is described by the equation F_tt12

^

K

&

~ L22d!sin~ F !1~dL₁2

^

K

&

2

22d2_{!sin~2F !}

12d

^

K

&

sin~3F !1d 2

2 sin~4F !50. ~43!

At derivation of this equation we take into account that F_t ;z, so we neglect by terms;zF_t. This equation is equiva-lent to a generalized pendulum motion whose effective po-tential is given

U~ F !522

^

K

&

~ L22d!212~ Ld12

^

K

&

222d2!cos~2F !

22d₃

^

K

&

cos~3F !2d 2

8 cos~4F !. ~44!

The additional minima at F₅₆np correspond to the trap-ping of oscillations in states with locked phase.

FIG. 9. The phase portrait in a planez,Fof the averaged system of equations~40!forF(0)5p, L52.0, andd50.2.

FIG. 10. The phase portrait in a plane z,F of the averaged system of equations~40!forF(0)5p, L52.0, andd50.6.

FIG. 11. The phase portrait in a plane z,F of the averaged system of equations~40!forF(0)5p, L52.0, andd50.8.

FIG. 12. The typical behavior of the solutions to Eq.~2!, ford

50.8 andL510,F(0)50 and varyingz(0). ~A! corresponds to

z(0)50.5,~B! toz(0)50.65,~C! to z(0)50.68, and~D! toz(0)

ization of atomic population should be modified in compari-son with the one considered earlier.

Let us find the fixed points of the averaged system ~40!. The first group is for F_n5pn, n50,61,62, . . . . For each F_n there can exist one or three roots forz. One of the roots isz150, two others are

zs1,256

S

12 4

^

K

&

2 ~ L2d!2

D

1/2

. ~45!

Note that if (L₂d)

^

K

&

,0, then three roots exists for evenn

and one root (z150) exists for odd n. For (L2d)

^

K

&

.0, one has the opposite situation. The critical value is L_c

.2

^

K

&

1d.

One can show, however, that linear stability analysis pre-dicts that this set is unstable whend_.L/2.

This root assumes thatu

^

K

&

/du,1. The linear stability analy-sis shows that the fixed points are unstable whenL_.3d.

Following @6#, from the constraint coming from the en-ergy conservation and the boundness of the tunneling enen-ergy, given by

H₅L1z

2_~0!

2 22

^^

K

&&

A1

2z

2_{~0!cos„F~0!…} 2d₂@1₂z2~0!#cos„2F~0!…>2

^

K

&

2d₂, ~48! we obtain the estimate for the critical value of L_cwhen the self-localization occurs. We find that

L_c54

^

K

&

@

A1

2z

2_~0!cosF~0!

11#₁d„@1₂z2~0!#cos 2F~0!₂1…

z2~0! 12d. ~49!

The phase portrait is plotted in Figs. 9–11 for initial phase difference F(0)₅p and L₅2.5, for different d 50.2,0.6,0.8. The regions corresponding to different dy-namics of z(t) with

^

z

&

50 and

^

z

&

Þ0 are shown. The in-creasing of d ~i.e., K1/V) leads to the distortion of the MQST regime, and for d;1 to the nonlinear Rabi-like os-cillations. The typical oscillations in time of the relative population for different values ofdand for fixedLare plot-ted in Fig. 12. The influence of rapid modulations leads to the appearance of new minima in the oscillations. This can be described as a result of the overlap of two double-well potentials. This conclusion is also confirmed by the results of numerical simulations of the GP equation for thesingleBEC under the rapidly varying trap potential performed in@18#. In this work, it has been shown that the effect of rapid modu-lations is the appearance of an effective double-well trap potential.

VII. CONCLUSION

In this paper we have studied the new effects coming the time variation of the trap potential and from damping on the nonlinear oscillations in the relative population behavior be-tween two BEC’s. For the slowly varying trap we predict synchronization of oscillations of the trap with oscillations of the relative population. We find the fixed points for both

types of damping terms occurring in the studies of two coupled condensates. Using the Melnikov approach, we study the possibility of the appearance of the chaotic oscil-lations in the tunneling phenomena between two coupled Bose-Einstein condensates. We calculate the width of the stochastic layer in phase space, in which the dynamics is chaotic. We find the lower bound on the region of chaos in the (K1,V) plane. We obtain the estimate for the damping coefficient when the chaos is suppressed. For the rapidly varying trap we use the multiscale method and derive the averaged equations for coupled modes describing the tunnel-ing phenomena. We find the fixed points in this case describ-ing the stationary states in two coupled BEC’s. The expres-sion for the critical value of the nonlinearity parameter, when the macroscopic quantum self-trapping occurs, is derived @26#. Our results show that there is interest to study the con-tribution of the quantum tunneling for this driven case and the investigation of resonances and chaos in oscillations of atomic population for the strongly overlapped condensates @27#. These problems require separate investigation.

ACKNOWLEDGMENTS

@1#M. R. Andrews, C. G. Townsend, and W. Ketterle, Science 275, 637~1997!.

@2#J. Javanainen, Phys. Rev. Lett.57, 3164~1986!.

@3#F. Dalfovo, L. Pitaevski, and S. Stringari, Phys. Rev. A 54, 4213~1996!.

@4#A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett.79, 4950~1997!.

@5#G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A55, 4318~1997!.

@6#S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A59, 620~1999!.

@7#I. Zapata, F. Sols, and A. Leggett, Phys. Rev. A 57, R28 ~1998!.

@8#I. Mazino, S. Raghavan, S. Fantoni, S. R. Shenoy, and A. Smerzi, Phys. Rev. A60, 487~1999!.

@9#A. Barone and B. Paterno, Physics and Applications of the Josephson Effect~Wiley, New York, 1982!.

@10#S. M. Jensen, IEEE J. Quantum Electron.18, 1580~1982!; A. A. Maier, Sov. J. Quantum Electron.12, 1490~1982!. @11#F. Kh. Abdullaev, S. A. Darmanyan, and P. K. Khabibullaev,

Optical Solitons~Springer, Heidelberg, 1993!.

@12#S. Trillo, S. Wabnitz, E. M. Wright, and G. Stegeman, Opt. Lett.13, 871~1988!.

@13#F. Kh. Abdullaev, R. M. Abrarov, and S. A. Darmanyan, Opt. Lett.14, 131~1989!.

@14#S. A. Darmanyan, Opt. Commun.90, 301~1992!.

@15#I. M. Skinner, G. D. Peng, and B. A. Malomed, Opt. Commun. 113, 493~1995!.

@16#Y. Castin and R. Dum, Phys. Rev. Lett.77, 5315~1996!. @17#J. J. Garsia-Ripoll, V. M. Perez-Garsia, and P. Torres, Phys.

Rev. Lett.89, 1715~1999!.

@18#R. Dum, A. Sampera, K.-A. Suominen, M. Brewczyk, M. Kus, K. Rzazewski, and M. Lewenstein, Phys. Rev. Lett.80, 3899 ~1998!.

@19#J. A. Dunningham and K. Burnett, Phys. Rev. Lett.82, 3729 ~1999!.

@20#J. Williams, R. Walser, J. Cooper, E. Cornell, and M. Holland, Phys. Rev. A59, R31~1999!.

@21#J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dy-namical Systems, and Bifurcations of Vector Fields ~Springer-Verlag, New York, 1983!.

@22#F. Kh. Abdullaev, Phys. Rep.179, 1~1989!.

@23#V. M. Kenkre and G. P. Tsironis, Phys. Rev. B 35, 1473 ~1987!.

@24#J. Kevorkian and J. D. Cole,Perturbation Methods in Applied Mathematics~Springer, New York, 1981!.

@25#T. S. Yang and W. L. Kath, Opt. Lett.22, 985~1997!. @26#F. Kh. Abdullaev,Theory of Solitons in Inhomogeneous Media

~Wiley, Chichester, 1994!.